Genus of quaternion algebras over formally real fields

Sergey V. Tikhonov

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arXiv · 2610.05392

Genus of quaternion algebras over formally real fields

Sergey V. Tikhonov

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Abstract

We prove the finiteness of the genus of quaternion division algebras over some formally real fields. In particular, if $F$ is a formally real field and the second power of the fundamental ideal $I^2(F)$ is torsion-free, then the genus of any quaternion division $F$-algebra is trivial. For instance, this immediately implies that the genus vanishes for any quaternion division algebra over a formally real pythagorean field. The proofs are based on methods from the theory of quadratic forms.

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BibTeX RIS

Sergey V. Tikhonov. 2026-10-04. Genus of quaternion algebras over formally real fields. https://arxiv.org/abs/2610.05392

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